Statement:
Every real number can be written as a difference of two positive real numbers.
My attempt:
$$ (\forall x\in\Bbb R)(\exists a\in\Bbb R)(\exists b\in\Bbb R)(x=a-b)\;$$
How do I specify that the two numbers are positive using correct notation?
Also, I am not quite sure how to express the following:
“There is a smallest positive real number.”
Every real number can be written as the difference of two positive real numbers:
$$\forall r\in\mathbb{R}, \exists a\in\mathbb{R}^+, \exists b\in\mathbb{R}^+ : r=a-b.$$ There is a smallest positive number (which is not true): $$ \exists s\in\mathbb{R}^+: \forall a\in\mathbb{R}^+, s\leq a$$ Not true since, according to the convention "positive means positive", i.e. $0\not\in\mathbb{R}^+$, that would imply $s\leq\frac{s}{2}$ and $1\leq \frac{1}{2}$ by the positivity of $s$.